CS 1050 B: Constructing Proofs April 7, 2009
Homework 7
Lecturer: Sasha Boldyreva Due: April 16, 2009
Do the assigned reading!
Assignment 7.01 Do the assigned reading.
Assignment 7.02 Indicate how much time did you spend on this homework.
In all problems do not just give an answer, but show your reasoning/work.
Problem 7.1, 5 points. In the proof of the Birthday paradox we used the following
very useful fact known as the union bound:
Pr(E1∪E2. . . En)≤Pr(E1) + Pr(E2) + . . . Pr(En),
where E1, E2,...Enare events associated with the same sample space. Prove it by induc-
tion and use the base case n= 2.
Problem 7.2, 5 points. Suppose that all you know about high tide is that its ex-
pected height is 1 meter, and the height is always non-negative. What can we say about
the probability that you see high tide more than 2 meters?
Problem 7.3, 5 points. Problem 16 from Section 6.4 of Rosen’s textbook.
Problem 7.4, 5 points. Problem 26 from Section 6.4 of Rosen’s textbook.
Problem 7.5, 5 points. Run the Extended GCD algorithm by hand on input (1529,14039)
and show the intermediate and final results.
Problem 7.6, 5 points. Prove that for any integers a, b, c, if c|(ab), and gcd(a, c) =
1, then c|b.
Problem 7.7, 5 points. Explain why adding two numbers aand bmodulo ntakes
linear time (in |n|), if 0 ≤a, b < n.
Problem 7.8, 5 points. Prove that if n is an integer that is not a multiple of 3, then
n2≡1mod 3.